Page:Scientific Papers of Josiah Willard Gibbs.djvu/268

be determined by means of the relation (500). But our measurements are practically confined to cases in which the difference of the pressures in the homogeneous masses is small; for with increasing differences of pressure the radii of curvature soon become too small for measurement. Therefore, although the equation $p'=p''$ (which is equivalent to an equation between $t,\mu _{1},\mu _{2}$, etc., since $p'$ and $p''$ are both functions of these variables) may not be exactly satisfied in cases in which it is convenient to measure the tension, yet this equation is so nearly satisfied in all the measurements of tension which we can make, that we must regard such measurements as simply establishing the values of $\sigma$ for values of $t,\mu _{1},\mu _{2}$, etc., which satisfy the equation $p'=p''$, but not as sufficient to establish the rate of change in the value of $\sigma$ for variations of $t,\mu _{1},\mu _{2}$, etc., which are inconsistent with the equation $p'=p''$.

To show this more distinctly, let $t,\mu _{2},m_{3}$, etc., remain constant, then by (508) and (98)

$d\sigma =-\Gamma _{1}d\mu _{1},$ $dp'=\gamma _{1}'d\mu _{1},$ $dp''=\gamma _{1}''d\mu _{1},$

$\gamma _{1}'$ and $\gamma _{1}''$ denoting the densities ${\frac {m_{1}'}{v'}}$ and ${\frac {m_{1}''}{v''}}\cdot$ Hence,

$dp'-dp''=(\gamma _{1}'-\gamma _{1}'')d\mu _{1},$

But by (500)

$(c_{1}+c_{2})d\sigma +\sigma d(c_{1}+c_{2})=d(p'-p'').$

Therefore,

$\Gamma _{1}(c_{1}+c_{2})d\sigma +\Gamma _{1}\sigma d(c_{1}+c_{2})=d(p'-p'').$

Now $\Gamma _{1}(c_{1}+c_{2})$ will generally be very small compared with $\gamma _{1}''-\gamma _{1}'.$ Neglecting the former term, we have

${\frac {d\sigma }{\sigma }}={\frac {\Gamma _{1}}{\gamma _{1}''-\gamma _{1}'}}d(c_{1}+c_{2}).$

To integrate this equation, we may regard $\Gamma _{1},\gamma _{1}',\gamma _{1}''$ as constant. This will give, as an approximate value,

$\log {\frac {\sigma }{\sigma '}}={\frac {\Gamma _{1}}{\gamma _{1}''-\gamma _{1}'}}(c_{1}+c_{2}),$

$\sigma '$ denoting the value of $\sigma$ when the surface is plane. From this it appears that when the radii of curvature have any sensible magnitude, the value of $\sigma$ will be sensibly the same as when the surface is plane and the temperature and all the potentials except one have the same values, unless the component for which the potential has